Scalable Hardware for Sparse Systems of Linear Equations, with Applications to Integer Factorization

Abstract

Motivated by the goal of factoring large integers using the Number Field Sieve, several special-purpose hardware designs have been recently proposed for solving large sparse systems of linear equations over finite fields using Wiedemann’s algorithm. However, in the context of factoring large (1024-bit) integers, these proposals were marginally practical due to the complexity of a wafer-scale design, or alternatively the difficulty of connecting smaller chips by a huge number of extremely fast interconnects.

In this paper we suggest a new special-purpose hardware device for the (block) Wiedemann algorithm, based on a pipelined systolic architecture reminiscent of the TWIRL device. The new architecture offers simpler chip layout and interconnections, improved efficiency, reduced cost, easy testability and greater flexibility in using the same hardware to solve sparse problems of widely varying sizes and densities. Our analysis indicates that standard fab technologies can be used in practice to carry out the linear algebra step of factoring 1024-bit RSA keys.

As part of our design but also of independent interest, we describe a new error-detection scheme adaptable to any implementation of Wiedemann’s algorithm. The new scheme can be used to detect computational errors with probability arbitrarily close to 1 and at negligible cost.